Calculus Examples

$f (x) = 4 x^{3} - 12 x^{2}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $4 x^{3} - 12 x^{2}$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 1.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x^{2}]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 (3 x^{2}) + \frac{d}{d x} [- 12 x^{2}]$

Step 1.1.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- 12 x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 12 x^{2}]$ .

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Step 1.1.3.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{2}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{2}]$ .

$12 x^{2} - 12 \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$12 x^{2} - 12 (2 x)$

Step 1.1.3.3

Multiply $2$ by $- 12$ .

$f' (x) = 12 x^{2} - 24 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 24 x$ .

$12 x^{2} - 24 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $12 x^{2} - 24 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$12 x^{2} - 24 x = 0$

Step 2.2

Factor $12 x$ out of $12 x^{2} - 24 x$ .

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Step 2.2.1

Factor $12 x$ out of $12 x^{2}$ .

$12 x (x) - 24 x = 0$

Step 2.2.2

Factor $12 x$ out of $- 24 x$ .

$12 x (x) + 12 x (- 2) = 0$

Step 2.2.3

Factor $12 x$ out of $12 x (x) + 12 x (- 2)$ .

$12 x (x - 2) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 2 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.6

The final solution is all the values that make $12 x (x - 2) = 0$ true.

$x = 0, 2$

Step 3

The values which make the derivative equal to $0$ are $0, 2$ .

$0, 2$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, 0) \cup (0, 2) \cup (2, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = 12 {(- 1)}^{2} - 24 \cdot - 1$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 1$ to the power of $2$ .

$f' (- 1) = 12 \cdot 1 - 24 \cdot - 1$

Step 5.2.1.2

Multiply $12$ by $1$ .

$f' (- 1) = 12 - 24 \cdot - 1$

Step 5.2.1.3

Multiply $- 24$ by $- 1$ .

$f' (- 1) = 12 + 24$

Step 5.2.2

Add $12$ and $24$ .

$f' (- 1) = 36$

Step 5.2.3

The final answer is $36$ .

$36$

Step 5.3

At $x = - 1$ the derivative is $36$ . Since this is positive, the function is increasing on $(- \infty, 0)$ .

Increasing on $(- \infty, 0)$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(0, 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = 12 {(1)}^{2} - 24 \cdot 1$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

One to any power is one.

$f' (1) = 12 \cdot 1 - 24 \cdot 1$

Step 6.2.1.2

Multiply $12$ by $1$ .

$f' (1) = 12 - 24 \cdot 1$

Step 6.2.1.3

Multiply $- 24$ by $1$ .

$f' (1) = 12 - 24$

Step 6.2.2

Subtract $24$ from $12$ .

$f' (1) = - 12$

Step 6.2.3

The final answer is $- 12$ .

$- 12$

Step 6.3

At $x = 1$ the derivative is $- 12$ . Since this is negative, the function is decreasing on $(0, 2)$ .

Decreasing on $(0, 2)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(2, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = 12 {(3)}^{2} - 24 \cdot 3$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $3$ to the power of $2$ .

$f' (3) = 12 \cdot 9 - 24 \cdot 3$

Step 7.2.1.2

Multiply $12$ by $9$ .

$f' (3) = 108 - 24 \cdot 3$

Step 7.2.1.3

Multiply $- 24$ by $3$ .

$f' (3) = 108 - 72$

Step 7.2.2

Subtract $72$ from $108$ .

$f' (3) = 36$

Step 7.2.3

The final answer is $36$ .

$36$

Step 7.3

At $x = 3$ the derivative is $36$ . Since this is positive, the function is increasing on $(2, \infty)$ .

Increasing on $(2, \infty)$ since $f' (x) > 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, 0), (2, \infty)$

Decreasing on: $(0, 2)$

Step 9